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 A081568 Third binomial transform of Fibonacci(n+1). 7
 1, 4, 17, 75, 338, 1541, 7069, 32532, 149965, 691903, 3193706, 14745009, 68084297, 314394980, 1451837593, 6704518371, 30961415074, 142980203437, 660285858245, 3049218769908, 14081386948661, 65028302171639, 300302858766202 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Binomial transform of A081567. Case k=3 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2), a(0)=1, a(1)=k+1. a(n) = 4^n*a(n;1/4) = Sum_{k=0..n} binomial(n,k)(-1)^k F(k-1) 4^(n-k), which also implies Deléham's formula given below and where a(n;d), n=0,1,..., d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also Witula's et al. papers). - Roman Witula, Jul 12 2012 REFERENCES D. Chmiela, K. Kaczmarek, R. Witula, Binomials Transformation Formulae of Scaled Fibonacci Numbers, (submitted to Fibonacci Quart. 2012). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 R. Witula, Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009) 310-329, MR2555042 Index entries for linear recurrences with constant coefficients, signature (7,-11). FORMULA a(n) = 7*a(n-1) - 11*a(n-2), a(0)=1, a(1)=4. a(n) = (1/2 - sqrt(5)/10)*(7/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2)*(sqrt(5)/2 + 7/2)^n = A099453(n)-3*A099453(n-1). G.f.: (1-3*x)/(1-7*x+11*x^2). a(n) = Sum_{k=0..n} A094441(n,k)*3^k. - Philippe Deléham, Dec 14 2009 G.f.: Q(0,u)/x -1/x, where u=x/(1-3*x), Q(k,u) = 1 + u^2 + (k+2)*u - u*(k+1 + u)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013 MAPLE seq(coeff(series((1-3*x)/(1-7*x+11*x^2), x, n+1), x, n), n = 0 .. 30); # G. C. Greubel, Aug 12 2019 MATHEMATICA CoefficientList[Series[(1-3x)/(1 -7x +11x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 09 2013 *) LinearRecurrence[{7, -11}, {1, 4}, 30] (* Harvey P. Dale, Feb 01 2015 *) PROG (MAGMA) I:=[1, 4]; [n le 2 select I[n] else 7*Self(n-1)-11*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 09 2013 (PARI) Vec((1-3*x)/(1-7*x+11*x^2) + O(x^30)) \\ Altug Alkan, Dec 10 2015 (Sage) def A081568_list(prec):     P. = PowerSeriesRing(ZZ, prec)     return P((1-3*x)/(1-7*x+11*x^2)).list() A081568_list(30) # G. C. Greubel, Aug 12 2019 (GAP) a:=[1, 4];; for n in [3..30] do a[n]:=7*a[n-1]-11*a[n-2]; od; a; # G. C. Greubel, Aug 12 2019 CROSSREFS Cf. A000045, A161731 (INVERT transform), A007582 (INVERTi transform), A081569 (binomial transform). Sequence in context: A227504 A218984 A289800 * A026378 A265680 A255714 Adjacent sequences:  A081565 A081566 A081567 * A081569 A081570 A081571 KEYWORD easy,nonn AUTHOR Paul Barry, Mar 22 2003 STATUS approved

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Last modified October 15 15:14 EDT 2019. Contains 328030 sequences. (Running on oeis4.)